It looks like long division. First divide the number 15.000000... into pairs of digits moving left and right from the decimal point: |15|.00|00|00|...|00|. You can draw a line over the top as you might do in long division where you will be placing the answer. You then want the nearest integer square to 15 which is lower than it. 9 is the nearest (16 is too big). Write 3 over 15 in the answer and follow it by the decimal point. Write 9 under 15, subtract 9 from 15 and write the result under the 9. Bring down the pair of zeroes, making 600. Double 3 in the answer, making 6, and write it to the left as you would write a divisor in long division. You are now looking for a number to go with 6 so as to make a number between 60 and 69. The digit you add will also be used to multiply so that the result is as close to 600 (the "remainder" so far) as you can make but it must be less than 600. 8 fits this requirement because 68*8=544. Subtract 544 from 600 leaving 56 and bring down the next pair of zeroes making 5600. Write 8 in the answer over the first pair of zeroes, following the decimal point. The answer so far: 3.8.
Double the answer ignoring the decimal point (76). We're now looking for a digit to tack on to 76 that will produce a number less than or equal to the remainder 5600. 7 fits this requirement: 767*7=5369. The answer so far is 3.87. Subtract from 5600=231. Bring down the next pair of zeroes=23100. Double the answer as before=2*387=774 and continue to find the next digit, which is 2, because 3 would give 23229. Continue until you reach the required number of decimal places, remembering for every pair of zeroes you bring down there is only one digit on the answer.